روش پرتفوی برای مسئله فروش چند محصولی پسر روزنامه با قید بودجه

This paper investigates a portfolio approach to multi-product newsboy problem with budget constraint, in which the procurement strategy for each newsboy product is designed as portfolio contract. A portfolio contract consists of a fixed-price contract and an option contract. We model the problem as a profit-maximization model, and propose an efficient solution procedure after investigating the structural properties of the model. We conduct numerical studies to show the efficiency of the proposed solution procedure, and to compare three models with different procurement contracts, i.e., fixed-price contract, option contract, and portfolio contract. Numerical results are shown to demonstrate the advantage of the portfolio model, and sensitivity analysis is provided for obtaining some managerial insights.

Multi-product newsboy problem with budget constraint is a classical inventory management problem, which was firstly introduced by Hadley and Whitin (1963). After Hadley and Whitin’s seminal work, many researchers have investigated different models and solution methods for multi-product newsboy problems. Khouja (1999) presented a good literature review on these research. Due to the difficulty for solving large-scale problems, most recent works have focused on developing efficient solution methods, e.g., Lau and Lau, 1996, Erlebacher, 2000, Vairaktarakis, 2000, Moon and Silver, 2000 and Abdel-Malek et al., 2004. To address non-negativity constraints of the order quantities, Abdel-Malek and Montanari (2005) proposed a modified Lagrangian based method by analyzing the solution space. Zhang, Xu, and Hua (2009) provided an exact solution method for the problem with any continuous demand distribution. Zhang and Hua (2008) proposed a unified algorithm for solving a class of convex separable nonlinear knapsack problems, which include the singly constrained multi-product newsboy problem with box constraints. Zhang and Du (2010) studied the multi-product newsboy problem with limited capacity and outsourcing. In the classical newsboy problem, the product is procured from the supplier with fixed-price contract. Under this procurement strategy, the retailer will undertake the salvage loss resulting from lower realized demand. To avoid this risk, the retailer always does not order enough inventories to maximize the supply chain’s total profit under the fixed-price contract (Cachon, 2003). In order to maximize the supply chain’s total profit, and share the risk raised from demand uncertainty with supply chain partners, some different contract types are used for encouraging the retailer to increase the order in supply chain management practice, such as buy back contracts, revenue sharing contracts, quantity flexibility contracts, sales rebate contracts and quantity discount contracts (Cachon, 2003). These contracts are labeled as “flexibility contract”, in which a fixed amount of supply is determined when the contract is signed, but the amount to be delivered and paid for can differ from the quantity determined upon signing the contract. In comparison with fixed-price contracts, these flexibility contracts not only coordinate the supply chain, but also have sufficient flexibility (by adjusting parameters) to allow for any division of the supply chain’s profit between suppliers and retailers. For more details of these flexibility contracts, please refer to Cachon (2003). Option contract is one type of flexibility contracts (Martı´nez de Albéniz & Simchi-Levi, 2005), which is defined as an agreement between the retailer and the supplier, in which the retailer pre-pays a reservation cost up-front for a commitment from the supplier to reserve certain order quantity. If the retailer does not execute the option, the initial payment is lost. With option contract, the retailer can purchase any amount of supply up to the option reservation level by paying an execution cost for each unit purchased. In other words, option contract provides the retailer with flexibility to adjust order quantity depending on the realized demand, and hence the inventory risk can be lowered for the retailer by utilizing the flexibility of option contract. There are mainly two branches for the research on option contracts in supply chain management literature: one perspective is supply chain coordination with option contracts, e.g., Barnes-Schuster et al., 2002, Wu et al., 2002, Kleindorfer and Wu, 2003, Wu and Kleindorfer, 2005, Wang and Liu, 2007 and GomezPadilla and Mishina, 2009, etc.; the other is on a single firm’s optimal procurement decisions given particular contractual terms, e.g., Cohen and Agrawal, 1999, Marquez and Blanchar, 2004, Wang and Tsao, 2006 and Boeckem and Schiller, 2008, etc. In these research, some combinations of different contracts, such as fixed-price contract and option contract, have been investigated. In addition, some research on option contracts also took into account spot market since it is another source of supply for commodity products, e.g., Martı´nez de Albéniz and Simchi-Levi, 2005, Fu et al., 2006 and Aggarwal and Ganeshan, 2007. Spot market is a supply market in which products are sold for cash and delivered immediately. Contracts bought and sold on spot market are immediately effective. For some products, spot market can be used by the firm to purchase at any time; however, the product price on spot market is random. Over the last years, the emergence of the business-to-business trading exchange has transformed the procurement strategies, which provides spot market where buyers and sellers can trade products any time at online markets (Aggarwal & Ganeshan, 2007). As Carbone (2001) reported, 50% of Hewlett–Packard’s procurement cost was spent on fixed-price contract, 35% in option contracts, and the remaining was left to the spot market. Up to now, all the existing works on the combination strategies of different contracts focused on single product setting. We have not found any research on multi-product demand management with the combination of fixed-price contract and option contract. In this paper, we introduce a portfolio approach for managing multi-product newsboy problem with budget constraint, in which each product can be procured with a portfolio contract consisting of a fixed-price contract and an option contract. The dual contracts for each product in the problem make the optimal ordering decisions more challenging in multi-product setting. On one hand, the use of option contract for lowering the overage cost should be properly balanced against the additional cost of using the option contract since unit reservation plus execution cost of option contract is typically higher than unit cost of a fixed-price contract. On the other hand, the total budget should be well allocated to different products for signing the fixed-price contracts and option contracts. The overall objective of the newsboy is to decide the optimal quantities of portfolio contracts for maximizing the total expected profit. We establish the structural properties for the optimal decisions of the proposed profit-maximization model, and develop an efficient solution procedure for the studied problem. Numerical results are shown to demonstrate the advantage of the portfolio model, and sensitivity analysis is provided for obtaining some managerial insights. The rest of the paper is organized as follows: Section 2 describes the problem formulation. In section 3, the properties of the optimal solution are established, and an exact solution procedure is developed. Section 4 is dedicated to numerical studies for demonstrating the advantage of the portfolio contract model, and obtaining some managerial insights from sensitivity analysis. Section 5 briefly concludes the paper and provides some future research directions.

In this paper, we investigate a portfolio approach to multi-product newsboy problem with budget constraint, in which the procurement strategy for the newsboy products is designed as portfolio contract. By establishing the structural properties of optimal solution, we develop an efficient solution method for the studied problem. The proposed algorithm has two main advantages: (1) it has linear computation complexity; (2) it is applicable to general continuous demand distribution. In comparison with fixed-price contract and option contract models, the portfolio contract model generates significant improvement when managing multi-product newsboy problem with budget constraint. Through sensitivity analysis, we come to the following insights: (1) the performance difference between fixed-price contract and portfolio contract models will become smaller as the available budget increases; (2) the performance gap between option contract and portfolio contract models increases when the available budget becomes larger. These insights suggest that managers with large budget should pay more attention to fixed-price contract if the portfolio contract is not available, and that managers with small budget should attempt to seek option contract if the portfolio contract cannot be used. There are several ways to extend this research. At first, this work can be directly extended to consider the scenario where different procurement strategies are available for different products. Secondly, another area of the future research is the consideration of an environment with supply uncertainty in sourcing and to investigate the effect of portfolio contract on managing supply uncertainty. Finally, one extension of this paper might be to study portfolio contract model with demand updating in multi-stage settings.